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\begin{center}\begin{large}\textbf{Math 137 - Problem Set 11\\
Due Wednesday, Apr 29}\end{large}\end{center}



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All rings are commutative, and $k$ is an algebraically closed field.

\begin{enumerate} 

\item Show that $V = \A^n-\{0\}$ is an affine variety if and only if $n = 1$. \footnote{Hint: what is $\Gamma(V)$?}



\item Let $C$ be the affine plane curve $V( x^3  + x^2 - y^2) \subset \A^2$. 

\begin{enumerate}

\item Show that $C$ is rational by explicitly describing a birational map $f: \A^1 \dashrightarrow C$ and checking that it induces an isomorphism of fields $f^*: k (C) \to k(\A^1) = k(t)$

\item Which DVR(s) in $k(t)$ dominate the local ring $f^*(\mathcal{O}_{(0,0)}(C))$?

\end{enumerate}


\item Let $X$ and $Y$ be varieties, and let $f: X \to Y$ be a function. Suppose $X = \cup_\alpha U_\alpha$ and $Y = \cup_\alpha V_\alpha$, where $U_\alpha \subset X$ and $V_\alpha \subset Y$ are open, and $f(U_\alpha) \subset V_\alpha$ for each $\alpha$.

\begin{enumerate}
\item Show that $f$ is a morphism if and only if each restriction $f_\alpha : U_\alpha \to V_\alpha$ of $f$ is a morphism. \footnote{Your answer here should not be super technical and complicated. The crux of this problem is just parsing definitions.}

\item If each $U_\alpha$ and $V_\alpha$ is an affine variety, show that $f$ is a morphism if and only if each $f^*(\Gamma(V_\alpha ) ) \subset \Gamma(U_\alpha)$. 
\end{enumerate}

\item Define a morphism $f :\P^1 \times \P^1 \to \P^{3}$ by 
$f([x_0: x_1], [y_0: y_1]) = [x_0 y_0 : x_0 y_1 : x_1 y_0 : x_1 y_1].$ This map is called the \textbf{\textit{Segre embedding}} of $\P^1 \times \P^1$ into $\P^3$.

\begin{enumerate}
\item Check that this map is well-defined and injective.
\item \textbf{(Extra credit)} Show that $f$ is a morphism.
\item Let the homogeneous coordinates of $\P^3$ be $t_{00}, t_{01}, t_{10},$ and $t_{11}$. Define  the ideal 
$$I = (t_{01}t_{10} - t_{00}t_{11}) \subset k[t_{00}, t_{01}, t_{10}, t_{11}].$$
 Show that the image of $f$ is $V(I)$.
\item \textbf{(Extra credit)} Generalize the above to $\P^n \times \P^m$. What is the ideal of the image of the map in this case?

\end{enumerate}

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